diff options
Diffstat (limited to 'EqualProps.thy')
| -rw-r--r-- | EqualProps.thy | 38 |
1 files changed, 28 insertions, 10 deletions
diff --git a/EqualProps.thy b/EqualProps.thy index 3b0de79..b691133 100644 --- a/EqualProps.thy +++ b/EqualProps.thy @@ -12,10 +12,11 @@ theory EqualProps Prod begin + section \<open>Symmetry / Path inverse\<close> definition inv :: "[Term, Term, Term] \<Rightarrow> Term" ("(1inv[_,/ _,/ _])") - where "inv[A,x,y] \<equiv> \<^bold>\<lambda>p: (x =\<^sub>A y). indEqual[A] (\<lambda>x y _. y =\<^sub>A x) (\<lambda>x. refl(x)) x y p" + where "inv[A,x,y] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. indEqual[A] (\<lambda>x y _. y =\<^sub>A x) (\<lambda>x. refl(x)) x y p" lemma inv_type: assumes "p : x =\<^sub>A y" @@ -57,21 +58,38 @@ proof - finally show "inv[A,a,a]`refl(a) \<equiv> refl(a)" . qed + section \<open>Transitivity / Path composition\<close> -\<comment> \<open>"Raw" composition function\<close> -definition compose' :: "Term \<Rightarrow> Term" ("(1compose''[_])") - where "compose'[A] \<equiv> - indEqual[A] (\<lambda>x y _. \<Prod>z:A. \<Prod>q: y =\<^sub>A z. x =\<^sub>A z) (indEqual[A](\<lambda>x z _. x =\<^sub>A z) (\<^bold>\<lambda>x:A. refl(x)))" +text "``Raw'' composition function, of type \<open>\<Prod>x,y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)\<close>." + +definition rcompose :: "Term \<Rightarrow> Term" ("(1rcompose[_])") + where "rcompose[A] \<equiv> \<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:x =\<^sub>A y. indEqual[A] + (\<lambda>x y _. \<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z) + (\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>p:x =\<^sub>A z. indEqual[A](\<lambda>x z _. x =\<^sub>A z) (\<lambda>x. refl(x)) x z p) + x y p" + +text "``Natural'' composition function abbreviation, effectively equivalent to a function of type \<open>\<Prod>x,y,z:A. x =\<^sub>A y \<rightarrow> y =\<^sub>A z \<rightarrow> x =\<^sub>A z\<close>." -\<comment> \<open>"Natural" composition function\<close> abbreviation compose :: "[Term, Term, Term, Term] \<Rightarrow> Term" ("(1compose[_,/ _,/ _,/ _])") - where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. \<^bold>\<lambda>q:y =\<^sub>A z. compose'[A]`x`y`p`z`q" + where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. \<^bold>\<lambda>q:y =\<^sub>A z. rcompose[A]`x`y`p`z`q" + -(**** GOOD CANDIDATE FOR AUTOMATION ****) lemma compose_comp: assumes "a : A" - shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" using assms Equal_intro[OF assms] unfolding compose'_def by simp + shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" + +proof (unfold rcompose_def) + have "compose[A,a,a,a]`refl(a) \<equiv> \<^bold>\<lambda>q:a =\<^sub>A a. rcompose[A]`a`a`refl(a)`a`q" + proof standard+ (*TODO: Set up the Simplifier to handle this proof at some point.*) + fix p q assume "p : a =\<^sub>A a" and "q : a =\<^sub>A a" + then show "rcompose[A]`a`a`p`a`q : a =\<^sub>A a" + proof (unfold rcompose_def) + have "(\<^bold>\<lambda>x:A. \<^bold>\<lambda>y:A. \<^bold>\<lambda>p:x =\<^sub>A y. (indEqual[A] + (\<lambda>x y _. \<Prod>z:A. y =[A] z \<rightarrow> x =[A] z) + (\<lambda>x. \<^bold>\<lambda>z:A. \<^bold>\<lambda>q:x =\<^sub>A z. (indEqual[A] (\<lambda>x z _. x =\<^sub>A z) refl x z q)) + x y p))`a`a`p`a`q \<equiv> ..." (*Okay really need to set up the Simplifier...*) +oops text "The above proof is a good candidate for proof automation; in particular we would like the system to be able to automatically find the conditions of the \<open>using\<close> clause in the proof. This would likely involve something like: @@ -81,7 +99,7 @@ This would likely involve something like: lemmas Equal_simps [simp] = inv_comp compose_comp -subsubsection \<open>Pretty printing\<close> +section \<open>Pretty printing\<close> abbreviation inv_pretty :: "[Term, Term, Term, Term] \<Rightarrow> Term" ("(1_\<^sup>-\<^sup>1[_, _, _])" 500) where "p\<^sup>-\<^sup>1[A,x,y] \<equiv> inv[A,x,y]`p" |